By selling 12 toffees for a rupee, a man loses 20%. How many for a rupee should he sell to get a gain of 20%?

### Concept & Formula When an item is sold at two different conditions, the constant factor is the Cost Price (CP). You can equate the ratio of Selling Price per unit to its corresponding percentage of CP. $$\frac{SP_1}{100 - \text{Loss } \%} = \frac{SP_2}{100 + \text{Gain } \%}$$ ### Step-by-Step Solution * **Given:** Case 1 SP = 12 toffees for ₹ 1. Loss = 20%. Case 2 SP = unknown toffees for ₹ 1. Gain = 20%. * **Calculation:** 1. SP of 1 toffee in Case 1 = $\text{₹ } \frac{1}{12}$. This is 80% (100% - 20%) of CP. 2. $CP \times 0.80 = \frac{1}{12} \implies CP = \frac{1}{12 \times 0.8} = \frac{1}{9.6} = \frac{10}{96} = \frac{5}{48}$ 3. Desired SP of 1 toffee for 20% gain = $CP \times 1.20 = \left(\frac{5}{48}\right) \times \left(\frac{120}{100}\right) = \left(\frac{5}{48}\right) \times \left(\frac{6}{5}\right) = \frac{1}{8}$ 4. If the SP is ₹ $\frac{1}{8}$ for 1 toffee, he should sell 8 toffees for ₹ 1. ### Exam Strategy & Shortcut Use the direct proportional formula linking Price, Quantity, and Percentage Value: $$\frac{\text{Amount}_1}{\text{Qty}_1 \times \%_1} = \frac{\text{Amount}_2}{\text{Qty}_2 \times \%_2}$$ $$\frac{1}{12 \times 80} = \frac{1}{x \times 120}$$ $$x = \frac{12 \times 80}{120} = \frac{960}{120} = 8$$ ### Common Pitfall A common mistake is trying to adjust the quantity (12) directly by the percentages without accounting for the inversely proportional relationship between quantity and price. ### Final Answer Therefore, the correct answer is **8**.